Unique properties of Smoothed Hankel functions

Smooth Hankel functions are convolutions of ordinary Hankel functions and Gaussian functions and are regular at the origin. Ordinary Hankel functions H¯L\bar H_L​H​¯​​​L​​ are solutions of the Helmholtz wave equation

Solutions are products of radial functions and spherical harmonics YL(r^)Y_L(\hat{\mathbf{r}})Y​L​​(​r​^​​), Here LLL is a compound index for the ℓm\ell{}mℓm quantum numbers. Radial functions are spherical Hankel or Bessel functions. We will focus on the Hankel functions:

where hℓ(1)(z)h_\ell^{(1)}(z)h​ℓ​(1)​​(z) is the spherical Hankel function of the first kind. Hankel (Bessel) functions are regular (irregular) as r→∞r \rightarrow \inftyr→∞ thus Hankel functions are exact solutions of the Schrödinger equation in a flat potential with appropriate boundary conditions for large rrr. For small rrr, the situation is reversed with Bessel functions being regular. Hankel and Bessel functions vary as r−ℓ−1r^{-\ell-1}r​−ℓ−1​​ and rℓr^{\ell}r​ℓ​​ when r→0r \rightarrow 0r→0.

When envelope functions are augmented with partial waves in spheres around atoms, the irregular part of H¯L\bar H_L​H​¯​​​L​​ is eliminated. Thus augmented Hankel functions can form exact solutions to the Schrödinger equation in a muffin-tin potential.

Smooth Hankel functions HLH_LH​L​​ are regular for both large and small rrr. The Figure below compares ordinary Hankel functions H¯L(ε,r)\bar H_L(\varepsilon,{\mathbf{r}})​H​¯​​​L​​(ε,r) (dashed lines) to smooth ones HLH_LH​L​​ for ε=−0.5\varepsilon{=}-0.5ε=−0.5. Red, green, and blue correspond to ℓ=0,1,2\ell=0,1,2ℓ=0,1,2.

Smooth Hankels are superior to ordinary ones, first because real potentials are not flat so there is scope for improvement on the Hankel functions as the basis set.

Also the fact that the HLH_LH​L​​ are everywhere smooth can greatly facilitate their implementation. In the present Questaal implementation the charge density is kept on a uniform mesh of points. Sharply peaked functions require finer meshes, and some smoothing would necessary in any case.

Finally also have a sensible asymptotic form, decaying exponentially as real wave functions do when far from an atom. Thus they have better shape than gaussian orbitals do.

Smooth Hankels have two big drawbacks as a basis set. First, they are more complicated to work with. One center expansions of ordinary Hankels (needed for augmentation) are Bessel functions. A counterpart does exist for smooth Hankels, but expansions are polynomials related to Laguerre polynomials. The expansion is cumbersom and introduces an extra cutoff in the polynomial order.

Second, gaussian orbitals hold an enormous advantage over both ordinary and smooth Hankels, namely that the product of two of them in real space can be expressed as another gaussian (plane waves have a similar property). There exist no counterpart for Hankels, so an auxiliary basis must be constructed to make the charge density and matrix elements of the potential. The Questaal suite uses plane waves for the auxiliary basis.

Smooth Hankel functions and the HkL family

Methfessel’s class of functions HkLH_{kL}H​kL​​, are a superset of smoothed Hankel functions HLH_LH​L​​; they also incorporate the family of (polynomial)×\times×(gaussians). The HLH_LH​L​​ and the HkLH_{kL}H​kL​​ are defined in reference 1, and many of their properties derived there. The HkL(r)H_{kL}({\mathbf{r}})H​kL​​(r) are a family of functions with k=0,1,2,...k=0,1,2,...k=0,1,2,... and angular momentum LLL. They are members of the general class of functions FL(r)F_L({\mathbf{r}})F​L​​(r) which are determined from a single radial function by

ΥL(r)\Upsilon_L({\mathbf{r}})Υ​L​​(r) with r=(x,y,z){\mathbf{r}}=(x,y,z)r=(x,y,z) is a polynomial in (x,y,z)(x,y,z)(x,y,z), so is meaningful to talk about ΥL(−∇)\Upsilon_L(-\nabla)Υ​L​​(−∇). It is written in terms of conventional spherical harmonics as

HLH_{L}H​L​​ is parameterized by energy ε\varepsilonε and smoothing radius rsr_sr​s​​; their significance will will become clear shortly. The extended family HkL(r)H_{kL}({\mathbf{r}})H​kL​​(r) is defined through powers of the Laplacian acting on HL(r)H_{L}({\mathbf{r}})H​L​​(r):

In real space HkLH_{kL}H​kL​​ must be generated recursively from hhh. However, the Fourier transform of HkLH_{kL}H​kL​​ has a closed form (Ref 1, Eq. 6.35). The differential operator becomes a multiplicative operator in the reciprocal space so

By taking limiting cases we can see the connection with familiar functions, and also the significance of parameters ε\varepsilonε and rsr_sr​s​​.

k=0k=0k=0 and rs=0r_s=0r​s​​=0: H^00(ε,0;q)=−4π/(ε−q2)\hat{H}_{00}(\varepsilon,0;\mathbf{q})=-{4\pi/(\varepsilon-q^2)}​H​^​​​00​​(ε,0;q)=−4π/(ε−q​2​​) This is the Fourier transform of H00(ε,0;r)=exp(−κ¯r)/rH_{00}(\varepsilon,0;r)=\exp(-{\bar\kappa}{}r)/rH​00​​(ε,0;r)=exp(−​κ​¯​​r)/r, and is proportional to the ℓ=0\ell=0ℓ=0 spherical Hankel function of the first kind, h0(1)(z)h_0^{(1)}(z)h​0​(1)​​(z). For general LLL the relation is

k=1k=1k=1 and ε=0\varepsilon=0ε=0: H^10(0,rs;q)=−4πe−rs2q2/4\hat{H}_{10}(0,r_s;\mathbf{q})=-{4\pi} e^{-r_s^2q^2/4}​H​^​​​10​​(0,r​s​​;q)=−4πe​−r​s​2​​q​2​​/4​​. This is the Fourier transform of a Gaussian function, whose width is defined by rsr_sr​s​​. For general LLL we can define the family of generalized Gaussian functions

Comparing cases 1 and 2 with Eq. (10), evidently H^L(q)\hat{H}_L(\mathbf{q})​H​^​​​L​​(q) is proportional to the product of the Fourier transforms of a conventional spherical Hankel function of the first kind, and a gaussian. By the convolution theorem, HL(r){H_L}({\mathbf{r}})H​L​​(r) is a convolution of a Hankel function and a gaussian. For r≫rsr\gg r_sr≫r​s​​, HL(r){H_L}({\mathbf{r}})H​L​​(r) behaves as a Hankel function and asymptotically tends to HL(r)→r−l−1exp(−−εr)YL(r^)H_L({\mathbf{r}})\to r^{-l-1}\exp(-\sqrt{-\varepsilon}r)Y_L(\hat{\mathbf{r}})H​L​​(r)→r​−l−1​​exp(−√​−ε​​​r)Y​L​​(​r​^​​). For r≪rsr\ll r_sr≪r​s​​ it has structure of a gaussian; it is therefore analytic and regular at the origin, varying as rlYL(r^)r^lY_L(\hat{\mathbf{r}})r​l​​Y​L​​(​r​^​​). Thus, the r−l−1r^{-l-1}r​−l−1​​ singularity of the Hankel function is smoothed out, with rsr_sr​s​​ determining the radius for transition from Gaussian-like to Hankel-like behavior. Thus, the smoothing radius rsr_sr​s​​ determines the smoothness of HLH_LH​L​​, and also the width of generalized gaussians GLG_LG​L​​.

By analogy with Eq. (9) we can extend the GLG_{L}G​L​​ family with the Laplacian operator:

The second equation shows that GkLG_{kL}G​kL​​ has the structure (polynomial of order kkk in r2r^2r​2​​)×GL\times G_L×G​L​​. These polynomials are related to the generalized Laguerre polynomials of half-integer order in r2r^2r​2​​. They obey a recurrence relation (see Ref 1, Eq. 5.19), which is how they are evaluated in practice. They are proportional to the polynomials PkLP_{kL}P​kL​​ used in one-center expansions of smoothed Hankels around remote sites (see Ref 1, Eq. 12.7).

Differential equation for smooth Hankel functions

Comparing the last form Eq. (14) to Eq. (10) and the definition of HkLH_{kL}H​kL​​ Eq. (9), we obtain the useful relations

This shows that HkLH_{kL}H​kL​​ is the solution to the Helmholz operator Δ+ε\Delta+\varepsilonΔ+ε in response to a source term smeared out in the form of a gaussian. A conventional Hankel function is the response to a point multipole at the origin (see Ref 1, Eq. 6.14). HkLH_{kL}H​kL​​ is also the solution to the Schrödinger equation for a potential that has an approximately gaussian dependence on rrr (Ref 1, Eq. 6.30).

Two-center integrals of smoothed Hankels

One extremely useful property of the HkLH_{kL}H​kL​​ is that the product of two of them, centered at different sites r1{\mathbf{r}}_1r​1​​ and r2{\mathbf{r}}_2r​2​​, can be integrated in closed form. The result a sum of other HkLH_{kL}H​kL​​, evaluated at the connecting vector r1−r2{\mathbf{r}}_1-{\mathbf{r}}_2r​1​​−r​2​​. This can be seen from the power theorem of Fourier transforms

and the fact that H^k1L1∗(q)H^k2L2(q)\hat{H}^*_{k_1L_1}(\mathbf{q})\hat{H}_{k_2L_2}(\mathbf{q})​H​^​​​k​1​​L​1​​​∗​​(q)​H​^​​​k​2​​L​2​​​​(q) can be expressed as a linear combination of other H^kL(q)\hat{H}_{kL}(\mathbf{q})​H​^​​​kL​​(q), or their energy derivatives. This is readily done from the identity

Comparing the first identity and the form Eq.~(10) of H^p0(q)\hat{H}_{p0}(\mathbf{q})​H​^​​​p0​​(q), it can be immediately seen that the product of two H^p0(q)\hat{H}_{p0}(\mathbf{q})​H​^​​​p0​​(q) with different energies can be expressed as a linear combination of two H^p0(q)\hat{H}_{p0}(\mathbf{q})​H​^​​​p0​​(q). The second identity applies when the H^p0(q)\hat{H}_{p0}(\mathbf{q})​H​^​​​p0​​(q) have the same energy; the product will involve the energy derivative of some H^p0(q)\hat{H}_{p0}(\mathbf{q})​H​^​​​p0​​(q). For higher LLL, ΥL1∗(−iq)ΥL2(−iq)\Upsilon^*_{L_1}(-i\mathbf{q})\Upsilon_{L_2}(-i\mathbf{q})Υ​L​1​​​∗​​(−iq)Υ​L​2​​​​(−iq) is expanded as a linear combination of ΥM∗(−iq)\Upsilon^*_{M}(-i\mathbf{q})Υ​M​∗​​(−iq) using the expansion theorem for spherical harmonics, Eq. (5). In detail,

When ε=0\varepsilon=0ε=0 and k≥1k\ge 1k≥1 the HkLH_{kL}H​kL​​ are generalized Gaussian functions of the type Eq. (12), scaled by −4π-4\pi−4π; see Eq. (16). Eq. (24) is then suitable for two-center integrals of generalized Gaussian functions.

Smoothed Hankels for positive energy

The smooth Hankel functions defined in Ref. 1 for negative energy also apply for positive energy. We demonstrate that here, and show that the difference between the conventional and smooth Hankel functions are real functions.

Other Resources

Many mathematical properties of smoothed Hankel functions and the HkLH_{kL}H​kL​​ family are described in this paper: E. Bott, M. Methfessel, W. Krabs, and P. C. Schmid, Nonsingular Hankel functions as a new basis for electronic structure calculations, J. Math. Phys. 39, 3393 (1998)